scholarly journals A Concise Derivation of Membrane Theory from Three-Dimensional Nonlinear Elasticity

2009 ◽  
Vol 97 (1) ◽  
pp. 97-101 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional ◽  
Membrane Theory

Membrane theory

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Three Dimensional ◽  
Thin Sheets ◽  
Two Dimensional ◽  
Leading Order ◽  
Membrane Theory ◽  
Dimensional Theory ◽  
Small Thickness

This chapter develops two-dimensional membrane theory as a leading order small-thickness approximation to the three-dimensional theory for thin sheets. Applications to axisymmetric equilibria are developed in detail, and applied to describe the phenomenon of bulge propagation in cylinders.

scholarly journals Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence

2009 ◽  
Vol 139 (5) ◽  
pp. 1037-1070 ◽  
Author(s):  
Lucia Scardia
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional ◽  
Curved Beam ◽  
Rigorous Derivation ◽  
One Dimensional ◽  
Asymptotic Models ◽  
Linearized Elasticity ◽  
Curved Rods ◽  
Γ Convergence ◽  
The One

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity.

scholarly journals Nonlinear elasticity in rocks: A comprehensive three-dimensional description

2017 ◽  
Vol 1 (2) ◽  
Author(s):  
Martin Lott ◽  
Marcel C. Remillieux ◽  
Vincent Garnier ◽  
Pierre-Yves Le Bas ◽  
T. J. Ulrich ◽  
...  
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional

scholarly journals A hierarchy of multilayered plate models

2020 ◽  
Author(s):  
Miguel de Benito Delgado ◽  
Bernd Schmidt
Keyword(s):  
Nonlinear Elasticity ◽  
Curvature Tensor ◽  
Elastic Constants ◽  
Three Dimensional ◽  
Spontaneous Curvature ◽  
Multilayered Plate ◽  
Von Karman ◽  
Plate Models ◽  
Γ Convergence ◽  
Von Kármán

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of Γ-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the Γ-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von Kármán, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.

Almost-everywhere injectivity in nonlinear elasticity

1988 ◽  
Vol 109 (1-2) ◽  
pp. 79-95 ◽  
Author(s):  
Tang Qi
Keyword(s):  
Weak Convergence ◽  
Nonlinear Elasticity ◽  
Partial Regularity ◽  
Three Dimensional ◽  
Inverse Function ◽  
Sufficient Condition ◽  
Almost Everywhere ◽  
Elasticity Problems ◽  
Image Position ◽  
Three Dimensional Elasticity

SynopsisThis paper gives a sufficient condition for almost-everywhere injectivity for nonlinear three dimensional elasticity similar to that of Claret-Necas [8], namely.We prove that this relation is maintained under the weak convergence of minimising sequences for nonlinear elasticity problems. The existence and partial regularity of an “inverse” function are proved.

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